Eq. (REF ) can be solved with Green's function method [1]}, [2]}. One of the two Green's functions satisfies the boundary condition,
\(g_{\widetilde{l}\widetilde{m}}^{(i+j)}\rightarrow {\left\lbrace \begin{array}{ll}\Delta ^{2}e^{-i\widetilde{k} r_{*}} & \text{ for }r\rightarrow r_{+},\\r^{3}B^{\mathrm {out}}_{\widetilde{l}\widetilde{m}}e^{i\widetilde{\omega }r_{*}}+r^{-1}B^{\mathrm {in}}_{\widetilde{l}\widetilde{m}}e^{-i\widetilde{\omega }r_{*}} & \text{ for }r\rightarrow +\infty .\end{array}\right.}\) 
